Functional Analysis and Infinite-Dimensional Geometry

Abstract

Library of Congress Cataloging-in-Publieation Data Funetional analysis andinfinite-dimensional geometry / Mariän Fabian . . . [et al.). p. cm . -(CMS books in mathematies ; 8) Ineludes bibliographical references andindex. I. Funetional analysis. 2. Banaeh spaces. I. Fabian, Mariän J. 1I. Series. QA320 .F793 2001 5I5'.7--de2 I 00-053773

Subspaces and quotients of Banach spaces with shrinking unconditional bases

“…The Enflo-James theorem (see [FHHMPZ,Theorem 9.18]) says that a Banach space X is superreflexive iff X is isomorphic to a uniformly convex space iff X is isomorphic to a uniformly smooth space.…”

Scalar Boundedness of Vector-Valued Functions

“…We use standard terminology and notation, which can be found in [3,6,17]. All our linear spaces are real.…”

Stability and Scenario Trees for Multistage Stochastic Programs

“…A σ-field F contains such a countable generator if it is generated by a R m -valued random vector. For these and related results we refer to [14,Sections 3 and 4]. Now, we are ready to state our existence result for solutions of (3).…”

“…For r ′ = 1 the compactness of any α-level set with respect to σ(L 1 , L ∞ ) follows from [36, Theorem 3K] due to condition (A5). For r ′ = ∞, some α-level set is bounded due to (A3) and, hence, relatively compact with respect to σ(L ∞ , L 1 ) due to Alaoglu's theorem [14,Theorem 3.21]. Since the objective function F (ξ, ·) is lower semicontinuous and N ∞ (ξ) weakly closed with respect to σ(L ∞ , L 1 ), the α-level set is even compact with respect to σ(L ∞ , L 1 ).…”

“…Since (ξ (n k ) ) norm converges in L r (Ω, F , P; R s ) to ξ and (x (n k ) ) weakly converges to x * , the sequence ((x (n k ) , ξ (n k ) )) of pairs in graph X converges weakly to (x * , ξ). Due to the closedness and convexity of graph X , Mazur's theorem [14,Theorem 3.19] implies that graph X is weakly closed and, thus, (x * , ξ) ∈ graph X or x * ∈ X (ξ). Finally, we have to show that x * belongs to N r ′ (ξ).…”